Numerical Approximation for Functionals of Reflecting Diffusion Processes

Abstract

The aim of this paper is to approximate the expectation of a large class of functionals of the solution (X,xi) of a stochastic differential equation with normal reflection in a piecewise smooth domain of Rd . This also yields a Monte Carlo method for solving partial differential problems of parabolic type with mixed boundary conditions. The approximation is based on a modified Euler scheme for the stochastic differential equation. The scheme can be driven by a sequence of bounded independently and identically distributed (i.i.d.) random variables, or, when the domain is convex, by a sequence of Gaussian i.i.d. random variables. The order of (weak) convergence for both cases is given. In the former case the order of convergence is 1/2, and it is shown to be exact by an example. In the last section numerical tests are presented. The behavior of the error as a function of the final time T, for fixed values of the discretization step, and as a function of the discretization step, for fixed values of the final time T, is analyzed.